Let Mod(S_g) be the mapping class group of the closed orientable surface S_g of genus g ≥ 2. In this paper, we derive necessary and sufficient conditions under which two torsion elements in Mod(S_g) will have conjugates that generate a non-abelian finite split metacyclic subgroup of Mod(S_g). As applications of the main result, we give a complete characterization of the finite dihedral and the generalized quaternionic subgroups of Mod(S_g) up to a certain equivalence that we will call weak conjugacy. Furthermore, we show that any finite-order mapping class whose corresponding orbifold is a sphere has a conjugate that lifts under certain finite-sheeted regular cyclic covers of S_g. Moreover, for g ≥ 5, we show the existence of an infinite dihedral subgroup of Mod(S_g) that is generated by an involution and a root of a bounding pair map of degree 3. Finally, we provide a complete classification of the weak conjugacy classes of the non-abelian finite split metacyclic subgroups of Mod(S₃) and Mod(S₅). We also describe nontrivial geometric realizations of some of these actions.

The first and third authors were supported by the UGC-JRF fellowship. The authors would also like to thank Dr. Siddhartha Sarkar for helpful discussions.